Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

In each of Problems 11-16, sketch the graph of the given Cartesian equation, and then find the polar equation for it. \(x-y=0\)

Step-by-Step Solution

Answer

The polar equation is \(\theta = \frac{\pi}{4} + n\pi\).

Step 1: Understand the Cartesian Equation

The given equation is a Cartesian equation: \(x - y = 0\). This equation can be rewritten as \(y = x\). This represents a straight line where every point has the same value of \(x\) and \(y\).

Step 2: Visualize the Cartesian Equation as a Graph

The equation \(y = x\) is a line with a slope of 1 that passes through the origin \((0,0)\). To sketch this, draw a line that makes a 45-degree angle with the positive \(x\)-axis, extending through the origin and continuing indefinitely in both directions.

Step 3: Convert Cartesian Coordinates to Polar Coordinates

In polar coordinates, a point is represented by distance from the origin \(r\) and an angle \(\theta\) from the positive \(x\)-axis. Recall the conversion formulas: \(x = r\cos\theta\) and \(y = r\sin\theta\).

Step 4: Substitute Values into Conversion Formulas

Substitute \(x = r\cos\theta\) and \(y = r\sin\theta\) into the equation \(x - y = 0\), leading to \(r\cos\theta - r\sin\theta = 0\).

Step 5: Simplify the Polar Equation

Factor out \(r\) from the equation \(r(\cos\theta - \sin\theta) = 0\). For non-trivial solutions, where \(r eq 0\), we need \(\cos\theta = \sin\theta\). This occurs when \(\theta = \frac{\pi}{4} + n\pi\) for any integer \(n\), which represents lines at 45-degree angles plus integer multiples of \(180\) degrees.

Key Concepts

Cartesian Equation

A Cartesian equation is a representation of a curve using the Cartesian coordinate system. In this system, the position of a point is described using two numbers, typically known as the x-coordinate and y-coordinate. For the equation \(x - y = 0\), it simplifies to \(y = x\). This linear equation signifies a straight line where each point on the line has equal x and y values.
Key characteristics of Cartesian equations include:

• They are typically expressed in the form \(ax + by = c\) where a, b, and c are constants.
• They describe a relationship or equation between x and y in a two-dimensional plane.
• The graph of a linear Cartesian equation is usually a straight line.

Cartesian equations are foundational in understanding more complex equations and forms, serving as a bridge to other coordinate systems like polar coordinates.

Graphing Equations

Graphing equations involve plotting points on a coordinate plane to represent an equation visually. The equation \(y = x\) translates to a line that passes through the origin with a slope of 1, forming a 45-degree angle with the x-axis.
When graphing, consider the background concepts:

• The slope of a line determines its steepness and direction. For \(y = x\), the slope (m) is 1.
• The y-intercept is the point where the line crosses the y-axis. For \(y = x\), this is at the origin (0,0).
• The graph continues indefinitely in both positive and negative directions along the x and y axes.

Graphing this equation helps visualize the straight line where every point on it can be linked back to the equation \(y = x\). This visual representation is essential for understanding the equivalence between the algebraic equation and its graphical form.

Coordinate Conversion

Coordinate conversion involves transforming points from one coordinate system to another. Here, we are converting Cartesian coordinates \((x, y)\) into polar coordinates \((r, \theta)\).
To achieve this conversion, use these formulas:

• \(x = r\cos\theta\)
• \(y = r\sin\theta\)

For the Cartesian equation \(x - y = 0\), substitute the polar conversions to get \(r\cos\theta - r\sin\theta = 0\). This simplifies down to \(\cos\theta = \sin\theta\), resulting in a condition for \(\theta\).
Understanding this conversion process is vital in finding the equivalent form of equations in different coordinate systems, facilitating analysis and application from different perspectives.

Polar Equation

A polar equation expresses a relationship in terms of the distance from the origin \(r\) and the angle \(\theta\) from the positive x-axis. These equations are particularly useful for problems involving circular or angular motion.
For the given Cartesian equation \(x - y = 0\), we found its polar equivalent by using coordinate conversion formulas. Substitute \(x = r\cos\theta\) and \(y = r\sin\theta\), leading to the polar equation \(\cos\theta = \sin\theta\), which simplifies to \(\theta = \frac{\pi}{4} + n\pi\), where \(n\) is any integer. This result corresponds to lines that pass through the origin making 45-degree angles.
Polar equations provide an alternative way to express relationships, particularly when symmetry and angular properties are involved. They can offer a more intuitive understanding of the rotational or radial aspects of a graph.